Quantum Mechanics

Description

Cover Contents Preface 1. Evolution of Physics: Classical to Quantum 1.1 Classical Mechanics 1.2 Light 1.3 Atoms 1.4 Some Views Based on Classical Physics 1.5 Black Body Radiation 1.6 Einstein’s Light Quantum 1.7 Dual Nature of Light: Wave–Particle Duality 1.8 Matter Waves 1.9 Vector Atom Model 1.10 Birth of Quantum Mechanics 1.11 Heisenberg’s Uncertainty Principle 1.12 Matter Wave: Experimental Evidences 1.13 Feynman’s Double-Slit Experiment: Exposition of Non-Classical Nature of Micro World 1.14 Particles and Waves from Quantum Field Theory Exercises References 2. Schrödinger Equation 2.1 Derivation of Schrödinger Equation 2.2 Hamiltonian Operator 2.3 Free-Particle Solution 2.4 Interpretation of Wave Function Ψ (r, t) 2.5 Normalization of Wave Function 2.6 Square Integrable Functions 2.7 Bound and Scattering States 2.8 Admissibility Conditions on a Wave Function in Quantum Mechanics 2.9 Conservation of Probability 2.10 Time-Independent Schrödinger Equation 2.11 Energy Eigenstates and Their Properties 2.12 Superposition of Energy Eigenstates – Time Evolution 2.13 Momentum Operator and its Eigenfunctions 2.14 Normalization of Momentum Eigenfunction 2.15 Coordinate and Momentum Representation for a System Appendix Exercises References 3. Simple Potentials 3.1 Scattering States 3.2 Bound State Solutions Appendix Exercises References 4. Mathematical Preliminaries 4.1 Linear Vector Space 4.2 Hilbert Space 4.3 Operators 4.4 Eigenvalue Equation 4.5 Dirac Delta Function Exercises References 5. General Formalism 5.1 Postulates 5.2 Commutation Relation between x and px 5.3 Compatible Observables 5.4 Uncertainty Principle 5.5 Complete Set of Compatible Observables 5.6 Constants of Motion 5.7 Ehrenfest Theorem 5.8 Classical Physics and Quantum Physics Exercises References 6. The Simple Harmonic Oscillator 6.1 Wave Mechanics 6.2 Abstract Operator Method 6.3 Coherent State Exercises References 7. Orbital Angular Momentum 7.1 Compatible Set of Angular Momentum Operators 7.2 Orbital Angular Momentum Operators in Spherical Polar Coordinates 7.3 Parity 7.4 Parity Operator in Spherical Polar Coordinates 7.5 Eigenvalues of L2 and Lz 7.6 Angular Momentum Eigen Functions 7.7 Spherical Harmonics as Eigenfunctions of Angular Momentum 7.8 Parity of Ylm(θ, ø) 7.9 Sketching of Spherical Harmonics 7.10 Raising and Lowering Operator 7.11 Eigenstates of Lx and Ly 7.12 Orbital Angular Momentum in Quantum and Classical Physics 7.13 Rigid Rotor Exercises References 8. Time-Independent Schrödinger Equation in Three Dimensions 8.1 Particle in a Rectangular Box 8.2 Harmonic Oscillator in Three Dimensions 8.3 Isotropic Harmonic Oscillator (Cartesian Coordinates) 8.4 Central Potentials: General Properties 8.5 Radial Equation 8.6 Centrifugal Potential 8.7 Radial Probability Distribution Function 8.8 Radial Momentum in Quantum Mechanics 8.9 Hamiltonian in Terms of Radial Momentum 8.10 Free Particle in Spherical Polar Coordinates 8.11 Particle in a Spherically Symmetric Box 8.12 Bound-State Solution for Spherically Symmetric Potential Well 8.13 Scattering State Solution for Square Well Potential 8.14 Two-Particle System in a Central Potential 8.15 Radial Equation for Coulomb Potential 8.16 Hydrogen Atom 8.17 Isotropic Harmonic Oscillator in Spherical Coordinates Appendix I Appendix II Exercises References 9. Bra and Ket Vector Formalism and Symmetries 9.1 Ket Vectors 9.2 Bra Vectors 9.3 Scalar Product 9.4 Abstract Operator 9.5 Adjoint of an Operator 9.6 Basis Vectors 9.7 Postulates of Quantum Mechanics 9.8 Representations 9.9 Coordinate Representation 9.10 Momentum Representation 9.11 Unitary Transformations 9.12 Time Evolution Operator 9.13 Space–Time Symmetries 9.14 Invariance of Hamiltonian 9.15 Active and Passive Transformations 9.16 Space Translation 9.17 Time Translation Symmetry 9.18 Rotational Invariance 9.19 Parity Invariance – Space Inversion 9.20 Time Reversal Exercises References 10. Angular Momentum and Rotation Symmetry 10.1 Eigenvalues of Ĵ2 and Ĵz 10.2 Matrix Representation of Angular Momentum Operators 10.3 Spin Angular Momentum 10.4 Spin 1/2 10.5 Pauli Matrices 10.6 Eigenvalues and Eigenvector of S·n 10.7 Rotation Matrix for Spin 1/2 Particle 10.8 Addition of Angular Momenta 10.9 Number of Basis Vectors in Coupled and Uncoupled Representation 10.10 Possible Values of J in Coupled Representation 10.11 Clebsh–Gordan Coefficients 10.12 Recursion Relations for Clebsh–Gordan Coefficient 10.13 Transformations of Operators Under Rotations 10.14 Spherical Tensor Operators Tq k 10.15 Wigner–Eckart Theorem Exercises References 11. Many-Particle Systems and Quantum Statistics 11.1 Identical Particles and Indistinguishability 11.2 Interchange Symmetry or Permutation Symmetry 11.3 Indistinguishability and the Interchange Operator 11.4 Symmetrization Postulate 11.5 Bosons – Construction of Symmetric Wave Functions 11.6 Fermion: Construction of Antisymmetric State 11.7 Pauli’s Exclusion Principle 11.8 Distinguishability as an Approximation 11.9 Two-Electron System Helium 11.10 Indistinguishability and Counting Microstate in Statitical Mechanics 11.11 Density Matrix 11.12 Pure State Density Matrix 11.13 Mixed State Density Matrix 11.14 Spin Density Matrix Exercises References 12. Electron in Magnetic Fields and Other Two-State Problems 12.1 Electrons in Classical Electromagnetic Field 12.2 Landau Energy Levels 12.3 Normal Zeeman Effect 12.4 Gauge Invariance of Schrödinger Equation 12.5 Aharonov–Bohm Effect 12.6 Two-State Problem 12.7 General Analysis of Two-State Problems 12.8 Magnetic Moment and Spin 12.9 Precession of Spin Angular Momentum 12.10 Magnetic Resonance 12.11 Ammonia Molecule 12.12 Neutrino Oscillation Exercises References 13. Time-Independent Schrödinger Equation – Approximations 13.1 Non-Degenerate Perturbation Theory 13.2 Degenerate Perturbation Theory 13.3 Stark Effect for Hydrogen Atom 13.4 Hydrogen Atom – Relativistic Effects and LS Coupling 13.5 Hydrogen Atom in Magnetic Field 13.6 Ground State of Helium—First-Order Perturbation Theory 13.7 Least Upper Bound for Ground-State Energy 13.8 Variational Method for Helium Atom 13.9 Slowly Varying Potential and WKB Approximation 13.10 WKB Expansion 13.11 Connection Formulae 13.12 Bohr-Sommerfield Quantization Rule 13.13 Energy Eigenvalues of Bound States Using WKB Approximation 13.14 Barrier Penetration 13.15 α Decay and WKB Approximation Exercises References 14. Time-Dependent Perturbation Theory 14.1 Time-Dependent Perturbation Theory–Basis 14.2 Transition Probability 14.3 First-Order Perturbation 14.4 Principle of Detailed Balancing 14.5 Constant Perturbation 14.6 Fermi’s Golden Rule 14.7 Harmonic Perturbation 14.8 Atoms and Radiation – Three Kinds of Transition 14.9 Atoms and Radiation – Semi-Classical Theory 14.10 Selection Rules 14.11 Einstein Coefficients 14.12 Adiabatic Approximation 14.13 Validity of Adiabatic Theorem 14.14 Adiabatic Theorem from Time-Dependent Perturbation 14.15 Berry Phase 14.16 Berry Phase for Rotating Magnetic Fields 14.17 Sudden Approximation Exercises References 15. The Scattering Theory 15.1 Scattering Cross Section 15.2 Scattering Amplitude and Differential Scattering Cross Section 15.3 Phase Shift Analysis 15.4 Scattering Amplitude and Phase Shift 15.5 Number of Partial Waves Needed 15.6 Integral Formula for Phase Shift δl 15.7 Expression for δl Using Logarithmic Derivative 15.8 Scattering by Hard Sphere 15.9 Phase Shift in the Low-Energy Approximation 15.10 Scattering Length in Low-Energy Scattering 15.11 Phase Shift for Square Well Potential 15.12 Breit–Wigner Formula 15.13 Green’s Function Technique in Non-Relativistic Scattering Theory 15.14 First-Order Born Approximation 15.15 Validity of Born Approximation 15.16 Born Approximation from Time-Dependent Perturbation Theory 15.17 Transformation for Scattering Cross Sections in Two Frames 15.18 Scattering Cross Section and Indistinguishability Appendix I Appendix II Exercises References 16. Relativistic Wave Equation 16.1 Klien Gordan Equation 16.2 Free Particle Solution to KG Equation 16.3 Problem with Negative Energy Solution 16.4 Probability Interpretation and KG Equation 16.5 Klien Gordan Equation and Hydrogen Atom 16.6 Motivations Leading to the Formulation of Dirac Equation 16.7 Meaning or Nature of α and β 16.8 Equation of Continuity 16.9 Solution to Free Particle Dirac Equation 16.10 Dirac’s Interpretation of Negative Energy States 16.11 Observable Consequences of Negative Energy Sea 16.12 Untenability of Interpretation of Ψ in the Non-Relativistic Sense 16.13 Spin Angular Momentum of an Electron 16.14 Helicity Operator 16.15 Magnetic Moment of an Electron Due to Spin Angular Momentum 16.16 Hydrogen Atom Exercises References Index